If you have a sequence a_n which is bounded below and a sequence b_n which is bounded below, show that a_n + b_n is bounded below. The definition of bounded below is a_n > -M.
your definition of being bounded below seems to be lacking. anyway, if that's what you are given, here is how we proceed. (i will write M instead of -M, 'cause that looks weird).
so we have $\displaystyle a_n > M$ and $\displaystyle b_n > K$ for all $\displaystyle n$.
thus, $\displaystyle a_n + b_n > M + K$ for all n. so $\displaystyle a_n + b_n$ is bounded below by M + K
um, it's not really something you have to prove. unless you want to go down to some kind of axiomatic level of number theory, which i doubt. it is just common sense that if you diminish each term, the result will be smaller
like since 2 > 1 and 3 > 1 it just makes sense that 2 + 3 > 1 + 1, because in place of two larger numbers, we put two smaller numbers, so the result must be smaller ...